Graphing Linear Inequalities

In the following diagram, all the points above the line y = 1 are represented by the inequality y > 1. All the points below the line are represented by the inequality y < 1. The representation is clearer if you look at what the y-coordinates of these points have in common.

In the diagram below, the region above the line is represented by y > 2x –1 and the region below the line is represented by y < 2x – 1.

Example:

By shading the wanted region, show the region represented by the inequality 2x – 3y ≥ 6

Solution:

First, we need to draw the line 2x – 3y = 6.

We will revise the method for drawing a straight line.

Rewrite the equation in the form y = mx + c.

From the equation m will be the gradient and c will be the y-intercept.

2x – 3y = 6

y = x – 2

The gradient is then and the y-intercept is – 2.

If the inequality is  or then we draw a solid line. If the inequality is < or > then we draw a dotted line.

After drawing the line, we need to shade the unwanted region.

Rewrite the inequality 2x – 3y ≥ 6 as y ≤ x – 2. Since the inequality is ≤ , the wanted region is below the line. We shade below the line.

Example:

By shading the unwanted region, show the region represented by the inequality x + y < 1

Solution:

Rewrite the equation x + y = 1in the form y = mx + c.

x + y = 1 can be written as
y = –x + 1

The gradient is then –1 and the y-intercept is 1.

We need to draw a dotted line because the inequality is <.

After drawing the dotted line, we need to shade the unwanted region.

Rewrite the inequality x + y < 1 as y < –x + 1. Since the inequality is < , the wanted region is below the line and so the unwanted region is above the line. We shade above the line.

Note: A question may require you to shade the wanted region or the unwanted region. If the question does not specify which region to shade, then state which region you are shading. While it may seem to be more intuitive to shade the wanted region here, shading the unwanted region is more useful in linear programming.

The following video shows an example of how to graph a linear inequality.

The video demonstrates how to graph linear inequalities.

Graphing Systems of Linear Inequalities

Example:

By shading the unwanted regions, show the region defined by the set of inequalities y < 2x + 5, y ≥ x , and x < 4.

Solution:

Draw the lines for each of the inequalities and shade the unwanted region for each of them.

The following videos show examples of how to graph systems of linear inequalities.

Graphing systems of inequalities

Solving Systems of Linear Inequalities

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Systems of Inequalities
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This tool will show the shaded region that satisfies all the given inequalities.

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